mixed_hes_fixed_obj#

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Hessian of Fixed Effects Objective#

Syntax#

   # mixed_obj.hes_fixed_obj(
      hes_fixed_obj_rcv    = None ,
      fixed_vec            = None ,
      random_opt           = None ,
   # )

Purpose#

We are given a value for the fixed effects \(\theta\) and the corresponding optimal value for the random effects \(\hat{u} ( \theta )\). This routine computes the hessian, with respect to the fixed effects, of the negative log of the Laplace approximation for the fixed effects objective

\[\B{p} ( z | \theta ) \B{p} ( \theta ) \int_{-\infty}^{+\infty} \B{p} ( y | \theta , u ) \B{p}( u | \theta ) \B{d} u\]

If there is no data, and not random effects, the return value is the Hessian of \(- \log [ \B{p} ( \theta ) ]\) .

hes_fixed_obj_rcv#

The argument hes_fixed_obj_rcv is a py_sparse_rcv matrix. The input value of this argument does not matter. Upon return it contains the lower triangle of the Hessian (the Hessian is symmetric).

fixed_vec#

The argument fixed_vec is a numpy vector with float elements and length n_fixed. It contains the value of the fixed effects \(\theta\) at which the Hessian is evaluated. This vector can’t be None.

random_opt#

The argument random_opt is a numpy vector with float elements and length n_random. It contains the optional value for the random effects, which is a function of the fixed effects and denoted by \(\hat{u} ( \theta )\) . This vector can’t be None.

Examples#